Research Plan

My current main project concerns the study of the interaction between the theory of operads and genuine equivariant homotopy theory. Briefly, an operad (introduced by May in [13]) consists of a sequence Oˆn of sets/spaces of “n-ary operations” together with Σn-actions and suitable compositions. The main point of operad theory is then the study of the algebras over a fixed operad O, which are objects X (in some appropriate monoidal category ˆC,a) together with n-ary operations suitably indexed by Oˆn. On the other hand, equivariant homotopy theory deals with the correct notion of homotopy when in the presence of the action of a group G. For example, a G-equivariant map f X Y between spaces with G-actions is considered a “genuine equivariant homotopy equivalence” only if f induces “non-equivariant equivalences” f X Y H between fixed points for all subgroups H B G. Work of Hill, Hopkins and Ravenel on the Kervaire invariant problem has revealed the importance of norms (given a G-object X and G-set A, the associated norm is the tensor product @AX together with an appropriate mixed G-action) and norm maps (i.e. maps between norms). Furthermore, follow up work of Blumberg and Hill in [1] has shown that (i) norm maps can be encoded in terms of G-equivariant operads by looking at certain fixed points Oˆn for special subgroups Γ B G Σn; (ii) general G-equivariant operads contain only some types of norm maps; (iii) each G-equivariant operad O has associated families of H-sets (for H ranging over subgroups H B G) which encode the norm maps present in O; (iv) these families of H-sets satisfy a number of novel and non-obvious closure conditions. Moreover, [1] calls families satisfying such closure conditions indexing systems and makes the key (and non-trivial) conjecture that all indexing systems can be realized by some operad. My current project stemmed from an attempt to find a conceptual understanding for the (somewhat opaque) closure conditions for indexing systems. In doing so, and jointly with Peter Bonventre, we discovered a theory of G-trees (a non-obvious generalization of the trees of Cisinski-Moerdijk-Weiss), which (i) provide a compact way to think about norm maps and their closure properties; (ii) suggest alternate models for equivariant operads.